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The subterranean nation of Glot has developed the ability to “thermally link” two equally sized diamonds, such that they are always the same temperature, regardless of spatial separation. We can consider this “thermal linking” process “magic” (or sci-fi "thermal quantum entanglement"), but physics otherwise operate the same as in our universe. Contrary to this particular advancement, Glot has NOT discovered electricity. Otherwise, it has technology equivalent to the early-mid 1800's.

One of the diamonds is carefully immersed in one of their many magma–rivers, the other is put into a steam engine boiler. (These magma-rivers are part of a very large underground area, and are accessible, with precautions, at minimal cost.) Always at equal temperatures, the two diamonds boil/heat the steam, and “cool” the magma river respectively.

The “linking” process is far less expensive than the diamonds themselves, so they are cut into slices, half a millimeter in thickness to increase surface area. These slices are the smallest they can mount, without having them “wash” away in the magma river, while keeping the diamond in contact with the magma.

Without considering the cost of the diamonds: would this process actually work to produce a useful, geothermal powered, steam engine? (That doesn’t require shoveling, or even carrying fuel?)

If so, what mass of diamonds, in total, would be required to power an average steam locomotive of the 19th century?

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    $\begingroup$ 0.5mm is really thin. Way too thin for 19th century technology. $\endgroup$ – RonJohn Oct 14 '18 at 11:05
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    $\begingroup$ Also a fascinating poison. Get a small paired diamond, peppercorn-size. Feed it to someone covertly. Then blackmail at your leisure: burn or freeze them from the inside if they don’t obey. Diamond may take a while to digest... and you can threaten to kill them immediately if they don’t eat another one. $\endgroup$ – SRM Oct 14 '18 at 14:31
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    $\begingroup$ More possibilities: Always-warm boots. Heated privy seats! Ice production in summer (just have someone with always-warm boots haul your diamonds to the Arctic)! Frost-protection for citrus groves! All subject to availability of diamonds, of course, so the wealthy may live VERY different lives from the poor! $\endgroup$ – SRM Oct 14 '18 at 14:36
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    $\begingroup$ How are you going to switch off your steam locomotives? For coal-fueled ones you just stop putting in more fuel, let it burn down, and cool out. You can't just keep it running dry, otherwise adding water again will cause an explosion. You will either need an elaborate system for raising the far counterparts from the magma at the right time, a quick disconnection and re-connection system, or you'll be limited to stationary long-running steam engines. $\endgroup$ – Bergi Oct 14 '18 at 16:44
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    $\begingroup$ @Pharap Diamond is an excellent conductor of heat. And a human hand on the receiving diamond would work -- sender moves candle under diamond, moves it away, back and forth. No, not as fast as electrical dots and dashes, but still allows long distance messaging. This telegraph could solve the signaling problem that Bergi raises -- tell the person manning the lava end to pull the diamonds out of the heat. $\endgroup$ – SRM Oct 14 '18 at 19:17
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The convection heat transfer coefficient for water heated by a solid surface, in a typical boiler, is about 1 kilowatt per square meter and kelvin. (That's why in a fire-heated boiler you have all those pipes, through which hot combustion gas flows; it it to maximize the surface area where the hot gas transfers heat to the water.)

This means that a shard of diamond with an area of 5×5 cm = 25 square centimeters (0.0025 square meters), 1000 K hotter than the water, will be able to transfer 2x2.5 = 5 kW. (I am assuming that the diamond is placed vertically so that both surfaces can transfer heat.) Locomotive steam engines have an efficiency of about 5% tops, so overall the engine will produce about 250 W, or 0.4 horsepower. A more-or-less typical steam locomotive produces about 2000 horsepower; it results that you need some 12.5 square meters of lava-temperature diamonds for this. (And you cannot have one big diamond, or else the heat transfer coefficient will come down abruptly; you need many small ones.)

The density of diamond is 3.5 grams per cubic centimeter. Assuming that the many small diamond pieces are 1 mm thick, the 12.5 square meters of heating come to 44 kg, or 220,000 carats. For uncut large diamonds, a price of 2,000 USD per carat is reasonable, so the 220,000 carats needed for one lousy steam locomotive would cost some 440 million USD. OK, maybe in your world De Beers doesn't exist so diamond prices are not massively inflated, but we are still speaking of something like 100 million USD.

Note 1: For the curious, I started with a 5x5 cm diamond because in my mind that's about the upper limit for an enormous diamond.

Note 2: I have no idea whether it is even possible to cut diamonds in thin flat slices. Since the calculations are crude approximations anyway, this shouldn't matter much.

Note 3: Lava is rarely hotter than 900° C, so a more natural temperature differential would have been 800 kelvin. I've taken 1000 kelvin for ease of calculation.

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    $\begingroup$ They could cleave the diamonds. $\endgroup$ – RonJohn Oct 14 '18 at 11:02
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    $\begingroup$ Caution: if you heat the diamonds above the heat transfer rate of the water, you may carbonize your diamond. $\endgroup$ – SRM Oct 14 '18 at 14:39
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    $\begingroup$ @SRM: Only the area in contact with water counts. You also want to keep the pieces large enough so that they can be somehow mounted in solid supports, because you don't want them to mix with the steam and go into the cylinders. $\endgroup$ – AlexP Oct 14 '18 at 15:56
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    $\begingroup$ @SRM the steam doesn't go up 'lighter than air'. It is under pressure, and then that pressure is released. Analogy: Carbon Dioxide in a Soda bottle, if the pressure is released quickly, more than just gas spews out the top. If the pressure is released slowly (ie low pressure) the engine isnt doing useful work. $\endgroup$ – Glen Davies Oct 14 '18 at 20:48
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    $\begingroup$ There is the small matter that if you cool the magma river too much, the magma will 'freeze' - like having to defrost a refrigerator, but with rock instead of ice. $\endgroup$ – Glen Davies Oct 14 '18 at 23:16
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Of course it would work. There's no practical difference between your diamonds and an electric heating element — except the diamond is experiencing 1,000℃ and the heating element usually isn't.

The issues are two-fold:

Surface Area: you need to get the heat off the diamond: more surface area = faster transfer.

Suspension: It would also be best to figure out how to suspend the diamond. Something that hot would eventually compromise the steel/iron/metal container it's sitting against (if it were simply thrown in), so your people would likely be experts at ceramics.

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